Journ@l Electronique d’Histoire des
Probabilités et de la Statistique
Electronic Journ@l for History of
Probability and Statistics
Vol 4, n°2; Décembre/December 2008
www.jehps.net
The Soul at the Razor’s Edge
J.-P. BENZECRI1
Translation from preprint version by Fionn Murtagh of L’Âme au
bout d’un rasoir , Les Cahiers de l’Analyse des Données, V (2),
229–242, 1980.
Intervening between logic and biology, a statistician who, first trained in
mathematics, went then towards linguistics and other human sciences, should
be cautious. But if he refrains both from overly technical formulas and overly
vague sentences what will remain for him to say? I will apply myself therefore
to raise, by the interplay of formulas, sufficiently suggestive images to illustrate,
animate and circumscribe a number of old maxims. Even if the authority of
erudition is no longer today met with as formerly among the scholarly in long
dress of Bologna or of Baghdad, I will go in quest of, sometimes far from us,
some learned testimonies to the universality of my subject.
1
Operations of the Spirit
On the threshold of his commentary on the De Interpretatione of Aristotle,
Saint Thomas recalls that according to the philosopher the operation of the
spirit is double: intelligence of the indivisibles (of that which every thing is in
itself); then formation of affirmative or negative propositions. To that a third
operation is added, reasoning, through which reason advances from the known
to the unknown. But is this not to run too fast? Turning around the edifice,
impeccable in itself, of the dialectic, Francis Bacon is ironical (Novum Organum,
I aphorism XIV): “The syllogism is made of propositions, the propositions of
words, and the words are the tokens of notions. Therefore if, at the base,
the notions themselves are confused, abstracted rashly from things, what one
constructs above is devoid of solidity”. And to conclude: Itaque spes est una in
inductione vera. “There is no hope except in a true induction.”
1 Professor
of Statistics, Université P. et M. Curie- Paris VI.
1
What is the illustrious Chancellor dreaming of? By a true induction, one
would define terms which would not be vain bags of wind but like networks
judiciously placed there where reality divides off from itself, and inserting in
their mesh an abundance of things: the interplay of words would then surely
mimic the interplay of these things. Thus, counter to what is too frequent
today, abstraction must not be understood at all as the state of that which
is autonomous (regulating itself according to its own laws, before dictating to
science), but as a process that without ever withdrawing from reality, takes its
forms from it. Assuredly if Bacon were to take up again amongst us the pen,
quickly wearying of scratching the invertebrate discourses of many an idealist
author who glorifies in perusing rhyme and reason, he would find in mathematics
and its logic, which are the new dialectic, worthy adversaries.
No tool, however – even if it would be this novius organum, so powerful
for induction, that is the electronic computer confronting a profusion of facts
through statistical calculation – would give to man a total direct hold on reality,
even if only practical. For want of being able to encompass, one takes with
tweezers: that suffices for the mysterious light of intelligence to touch being.
But without claiming, even were this planned, to make a clear sweep (and how
could one being placed as we are in the midst of things? never holding beginning
nor end; nor the infinitely small, nor the infinitely large; nor the part nor the
whole; nor our own thought no more than that of the world), without ceasing
to name, affirm, reason, as Aristotle saw it, one will allow us to turn back to
scrutinize definitions. All of biology, is it not in the single word of life?
2
Definition and Reasoning in Classical Logic
Let us recall, according to an excellent treatise on philosophy2 , what one means
classically in logic by extension and comprehension: this will be the occasion
then to contrast the thought of the biologist with contemporary formalist trends.
“One can”, writes R. Jolivet, “consider an idea and subsequently a term from
the point of view of comprehension and from the point of view of extension ...”.
1. Comprehension is the content of an idea, i.e. the set of elements of which
an idea is composed. Thus comprehension of the idea of man implies the
following elements: being, living, endowed with senses, reasonable.
2. Extension is the set of the subjects which the idea suits. It is thus that
the concept of “animal” of which the extension is larger (e.g. than that
of man) is called superior concept: the concepts which enter its extension
are its inferiors or its subjective parts.
The same author3 shows well how, from the point of view of the extension,
one obtains a sort of visual or geometric representation of the syllogism. It
2 Régis Jolivet, Traité de Philosophie (Treatise of Philosophy), Vol. 1, Section 46, 2nd edn.,
Vitte Lyon-Paris, 1945.
3 Ibid. Section 36.
2
Figure 1: The scheme of Euler symbolizing the extension of terms by three
concentric circles.
is thus that the schema of Euler symbolizes the extension of terms by three
concentric circles (Figure 1):
• Every man is mortal. Man (M) completely forms part of the extension of
Mortal (T).
• Now Pierre is a man. Pierre (t) forms part of the extension of Man (M).
• Hence Pierre is mortal. Pierre (t) forms part of the extension of Mortal
(T).
In this visual and geometric representation, the mathematician of today recognizes sets, and he is filled with wonder that in a time when set theory fever
did not yet burn the youngest foreheads the great name of Euler found a place
in the school manual of logic. Critiques came about:
“The disadvantages of the exclusive usage of this procedure are: first to
systematically materialize an essentially intellectual operation, lying on a vision
of the spirit and irreducible to any combination of figures, – then to get the spirit
used to a sort of “logic of classes” which, abandoning the objective content of
ideas, i.e. of objects of thought, does not consider them more than as empty
and purely arbitrary frames, susceptible to be replaced by some sign or other.
3
Instead of considering the type of logical necessity that links the conclusion
to the premises and that constitutes the essence of the reasoning, extensivist
logic ends by not considering any more the exterior sign which is supposed to
symbolize logical necessity, but which, in fact, conceals it from the spirit.”
All things considered, we subscribe to these critiques, ... in as much as we
understand them. What does logical necessity mean? We appreciate enough the
depth of thought of R. Jolivet to believe that with him these words are more
loaded with meaning than virtus dormitiva for Molière’s doctor. But, just as
an empty box resembles to the sight a full box, likewise the way of speaking of
philosophers often leaves a mathematician in doubt: What has he said? Did he
say anything? Moreover if a vision of the spirit is irreducible to any combination
of figures it is left that one cannot think without a figure:
νoιν oυκ στ ιν ανυ ϕαντ ασµατ oς (noein ouk estin aneu phantasmatos,
without an image present in the mind it is impossible to reason)
writes Aristotle4 . The object of the vision of the spirit is not a figure but
it is suggested by the concept of many images (as the concept of triangle by
many drawn triangles). Now mathematical language depicts with exactitude
and conciseness; we will make use of it to suggest the meaning of the critiques
formulated by R. Jolivet.
3
Logic and Formulas
Let us leave contemporary mathematical logic, where notions are very precise if
not perfectly appropriate for the thought of him who scrutinizes nature. Commonly, reasoning expresses itself in a language: before asking oneself if one is
telling the truth one can enquire as to whether one has just spoken sentences:
likewise mathematical logic presupposes a formal grammar. Fundamentally this
grammar is only a system of rules to decide among finite sequences of symbols
(taken in a set or alphabet decreed at the start) that are well-formed expressions,
and in particular propositions (or terms or sentences). Logic then considers
these propositions as being able to receive a value; and it provides these rules,
called deduction rules or schemas, to attribute a value to certain propositions,
the value of certain others being known. The reader, if he/she has guessed what
we are talking about, will get irritated perhaps that we have not been saying:
rules permitting to affirm that certain propositions are true or false, a certain
number of others being supposed true. This is because we are not yet where
we want to be: the set of possible values for a proposition is not necessarily the
set of two elements { true, false }; this can be a set of more than two elements,
to which in the formal interplay that is ours here one does not give a name.
Would one name them in as much as one would not have rejected the principle
of the excluded middle according to which it is not given to a proposition to
be other than true or false (a third outcome is excluded, tertium non datur);
because what we call here well-formed propositions are just assemblages of signs
4 On
Memory and Reminiscence 1, 4496.
4
to which it remains to give a meaning, a real referent; which has not yet been
done and which, once the correspondence rules decreed, is not perhaps feasible for every sentence (hence an example of the third outcome, for which the
philosopher would be wrong to blame the logician).
The correspondence between terms of a formal system and truths of a natural domain (e.g. biology) has hardly at all been scrutinized, for its own sake –
even in the detail. Around 1940 there was current an ambitious project, under
more than one great name, among them that of R. Carnap, for an international encyclopedia of unified knowledge (International Encyclopedia of Unified
Science). The ultimate objective was without doubt to start from elementary
truths of experience tabulated following the rules of a rigorous positivism, to
rise up by computational logic as far as the general terms of the sciences; but
only methodological volumes saw the light of day. In practice the logical description of the real passes through the mathematical description of the latter.
The formalists who do not appear to us to have had, on the appropriateness of
mathematics to the real, fertile nor profound views5 , on the other hand made
spectacular discoveries on what can be the appropriateness of mathematics of a
logical system of signs. Here is not the place to try only to present the works of
Kurt Gödel or of Paul Cohen; but to conceive of what becomes in contemporary
mathematical logic extension and comprehension, relations must be considered
between the statements of a formal system (or language of the logic) on the one
hand, and on the other hand the mathematical objects relative to which these
statements take on meaning; this is what we will now do briefly.
4
Syntax of Symbolic Logic
Up to now we have not said anything about the language where logical propositions are written. We wanted thus to emphasize that, for the logician, this
language is firstly an articulated interplay of signs, of which the rules can be
freely chosen, and not a simplified and perfectly regular translation of his/her
mother tongue. Having posed this principle, it remains that the variations of
language commonly met with, of one author to another, and even with a single
author, of one work to another, leave intact a common background of structure
which is not incomprehensible for an honest man practicing natural languages.
So much so that the very simple language that we present below offers to a true
logician everything necessary to pose the most profound problems.
The alphabet consists of the following 12 symbols:
z; &; ∨;
x; c; R; =;
∀;
(; ); ; ;
5 One should however credit the formalists with their contribution to the conception of
electronic computers; from which an indirect but in our view essential contribution to the
mathematical elaboration of real data; we will return to this point.
5
Rather than set out a formal grammar, let us suggest through correspondence
with common languages the rules of formation of propositions.
z is an adverb (or unary conjunction) which, placed before a proposition,
changes it into its negation.
& and ∨ are the binary conjunctions (“and” and “or”) that placed between
two propositions make of them a new proposition (that and this ...).
c (possibly assigned some number or other of raised accents: c , c , c , . . .)
designates a constant individual.
x (possibly assigned accents: x , x , . . .) designates an individual variable: a
variable is like an indefinite pronoun, all occurrences of which in the sentence
would be linked by a thread, similar to that which links the relative to its
antecedent: one knows thus that it is always a matter of the same indeterminate
individual. Natural languages are far from having this unlimited clarity.
R is a relational constant or predicate: if R is assigned n lower indices one
has an n-ary predicate (the number of raised indices is only a convenience that

ensures an infinite vocabulary): for example R
(x, c, x ) is a proposition that

expresses the relation R between three individuals x, c, x , of which two are
indeterminate.
= is a particular binary relation, equality.
∀ is the universal quantifier: ∀ x placed at the head of a complex proposition
(formed of predicates, conjunctions, variable or constant individuals) expresses
the possibility that the relation that expresses the proposition exists irrespective
of the particular individual taken for x.
Finally, on the way, we have noticed the use of parentheses and of accents.
Once the language has been described, it remains to give the principles or
schemas of deduction permitting to pass from a set of propositions or premises
supposed to have the value true to a conclusion of the same value. From these
schemas, which are the logic as such, we will limit ourselves to say that they
translate into formulas the, all in all, natural and classical rational principles: as
for deduction, contemporary formalism is not against the Aristotelian syllogism
(set out in the Prior Analytics) any more than modern algebra is not against the
arithmetic calculations of Diophantus. In our view, the neo-scholastic philosophers who are indignant at seeing the syllogism abandoned extend without due
cause to the entirety of mathematical logic a hostility that they should reserve
just for nominalism (professed it is true by a number of mathematicians; who,
we were recalling above, understand abstraction as a state, not as a process).
5
Relations and Models
Syntax and logic being fixed, one can constitute a particular theory as within
the framework of a natural language; one reasons from basic notions and truths.
The language described in the previous section contains two signs c and R
that, through the adding of accents, will furnish the theory with the vocabulary
that one desires. Sometimes one introduces infinite sets of individuals and of
predicates; but in fixing a finite number of them one has already matter for the
6
deepest research. With the vocabulary thereby decreed basic postulates of the
theory are written, which are the truth propositions from which one proposes
to deduce the logical consequences.
One must be careful even if the quantifier ∀ and the values (x, x , . . .) refer from meaning to the set of all possible individuals, this set (for which the
particular theory has only fixed some constant individuals c, c , c ) stays in the
dark. Logical computation itself operates without naming it and without even
postulating it. If we have spoken about it above it is to suggest the finality, the
possible utility of the interplay of formulas. To introduce such a set that we will
name I is to give a model to the theory. And it is not sufficient that I consists
of the constants (c, c , c ) used in the base vocabulary, truth relations must still
be specified for each predicate of this vocabulary: for example, if i, i , i are

particular individuals of the set I, one should say if R
(i, i , i ) is true or not;
finally the relations thus specified must, and this is essential, conform to the

premises of the theory (for example, if one has as premises: ∀x ∀x R
(x, x, x ),


one must have in particular R (i, i, i )).
Mathematical logic demonstrates the existence of a model and the noncontradiction of a theory of the theorems (due principally to K. Gödel) which are
of real philosophical impact: on the one hand, briefly, a model exists if and only
if the theory is non-contradictory (if one cannot demonstrate simultaneously
there a proposition A and its negation 1 A); on the other hand, to establish
that an even remotely rich theory (e.g. the arithmetic of whole numbers) is noncontradictory requires the help of an even richer theory (for which one can claim
even less that it is non-contradictory). Certain mathematicians conclude that
hypothetical mathematical beings must be the least possible postulated and they
give precise rules to their asceticism; this theory, which is that of intuitionism,
has very resolute adversaries. Without having anything profound to contribute
to this internal debate in mathematics we will suggest that the coherence of a
theory is best demonstrated by its appropriateness for natural phenomena of a
certain order: for, if the appropriateness is perfect, nature supplies a model to
the theory; and if, as it is likely, the appropriateness is only imperfect, one is
at least assured that the theory will be able, without disappearing, to improve
itself and eliminate its contradictions, becoming a better approximation of the
real. (To demonstrate within a theory A ∨ (1 A), A and non-A, can appear
an absolute vice; but this demonstration can result from a long chain, that a
distinction introduced into an axiom suffices to break.)
6
Notions in Formulas
With a view to this confrontation with the real, let us see from examples what
is in mathematical logic a notion, its extension, then its comprehension. First
let R be a unary predicate: the extension of R is fixed with the model: this is
the set of individuals for which one has specified that R (i) is true. But there
are in the theory more complex notions than the basic unary predicates; for
7
example the statement

∀xR (i, x) ∨ R
(i, x, x)

(i, x, x) is true) defines
(whatever the determined x either R (i, x) or else R
for i a notion N : the statement is true for individuals i of a certain subset of I,
that is by definition the extension of N and is commonly denoted:

I(N ) = {i|i ∈ I; ∀xR (i, x) ∨ R
(i, x, x)}
With every similar statement that includes, beyond predicates and a variable
x (there could be several of them) linked to the quantifier ∀, the symbol i of a
free individual (not linked to a quantifier) is associated a notion. Among the
parts of I, certain only can, like I(N ), be described as the extension of a notion
defined within the theory of which I is a model.
As for comprehension, let us consider first a notion N  defined by the conjunction of three unary predicates, R , R , R , and for which the extension
is
I(N  ) = {i|i ∈ I; R (i) ∧ R (i) ∧ R (i)} ;
(i.e. the set of individuals for which R (i) , R (i) , R (i) are simultaneously
true). It is natural to say that the comprehension of this notion is the set of
three predicates {R ,R ,R }. But in proceeding thus it is impossible to define
every notion by its comprehension. It is more correct to say that on the one
hand the notions, because they are defined by means of basic predicates (or
properties), are in a certain sense defined as comprehension rather than as
extension; and on the other hand that the relations between notions (e.g. that
N  is a consequence of N ) can be expressed not only in extension, in the context
of a model (I(N ) ⊂ I(N  ), every i that possesses N possesses N  ) but also
independently of every model, as a proposition of the theory (which will be for
N and N  ):

(x , x, x)) ∨ 1(R (x ) ∧ R (x ) ∧ R (x )) ,
∀x (∀xR (x , x) ∨ R
that is, that every x either possesses N  or does not possess N .
If one accepts this point of view, formal logic works far more as comprehension than as extension and the philosophical critiques of authors such as R.
Jolivet are not at all about that. But one must be careful in that being opposed
to a logic of classes founded on extension, these philosophers wish to prevent
the spirit to stop at symbols while losing sight of the objective content of ideas,
the content which alone according to them founds logical necessity. By objective
content, it must be understood these domains of real facts, which the notions do
not lock up perfectly enough for the articulations to be able to be enumerated in
axioms; logical necessity results finally always from a to and fro between things
and words: the spirit seizes things by words and calculates on the latter; but
it has to test if the separation between things and words (linked by a more or
less narrow, but non-identified, analogy) does not render illusory to affirm from
the former the conclusions obtained on the latter. On natural relations between
facts and concepts we will speak now.
8
7
Inductive Definition
At the threshold of an Introduction6 to biology destined for university students,
our colleague Professor P. Favard writes: “Rather than making abstract reconstruction of an ideal cell, we will describe the organisation of cells belonging to
very dissimilar organisms ... For in their infinite diversity of organisation, the
cells are only, all things considered, variations on a single theme ...”. And to
propose a series of examples; whether of cells incorporated in a pluricellular being, animal or vegetal; or of unicellular organisms – protozoa, bacteria or green
algae.
Thus for P. Favard, the idea of cell is not a model, a type that the abstraction
would be able to fix; it is that which several particular living entities have as
analogs. The idea is not conceived if the spirit does not move from the particular
to the general; to this inductive movement the author prompts the reader. Not
only the abstraction conceived as the state of that which is separated from the
real is vain but more the result of the process of abstraction is elusive: only the
process itself holds.
When, as is the case here for the cell, a notion is defined as a polygonal zone
underpinned by some individuals, we will speak of an inductive definition. We
will see with an example that multidimensional statistical analysis can offer the
inductive definition a rigorous foundation in giving to our image of the polygonal
zone a concrete realization in a plane or a space.
Let us propose now with S. Blaise to study the systematics of the genus
Myosotis7 .
“This genus of great morphological homogeneity includes a high enough
number of species and of sub-species all the more difficult to discriminate on
the one hand since there exist hybrids (intermediary forms) and since on the
other hand the same vegetal individual with determined genetic capital can, in
accordance with the conditions of its growth, develop strongly diverse forms.
... The determination of a base of Myosotis rests on the descriptive characters
which we will arrange in three classes:
A) Apparent qualitative characters: such are the color and the form of the
corolla, the pilosity of the leaves, etc.
B) Architectural qualitative characters, easily quantified: we will cite the distance
between two nodes, the number of leaves on such a branch ...
C) Microscopic characters: these are palynological characters (form of pollen)
and karyological (number of chromosomes ...) ...
The set of these characters allows very sure determinations. ... But in
daily botanical practice, the microscopic characters C) are not known; the floral
characters A) cited in ancient flora are often unusable in the state considered of
development and of conservation of the base. Hence the interest of a statistical
6 cf.
M. Durand and P. Favard: La Cellule (The Cell), Hermann, Paris (1967)
S. Blaise, J.-P. Briane and M.-O. Lebeaux, Le genre Myosotis, in J.-P. Benzécri et al.,
L’Analyse des Données, Vol. I, C no. 1, Dunod, 1973.
7 cf.
9
study of the architectural macroscopic characters B), the most general and the
most easily determined.”
Let us sketch this study. Suppose that one considers 13 macroscopic characters; with every base of Myosotis is thus associated a system of 13 numbers, that
is, in geometric language, a point of the space R13 with 13 dimensions; and with
the set of Myosotis of every species that one has measured there corresponds a
cloud of points of R13 . In this space however the eye of man has seen nothing!
But statistical computation, for which the computer is the indispensable tool,
provides generally for a cloud of R13 a satisfactory map in the plane, indeed in
the usual three-dimensional space; (we say generally because the cloud could be
too complex, the density of the set of possible forms might not be expressed in 2
or 3 dimensions; this is not the case here). On the planar map one notices first
that the two large groups of Myosotis that one distinguishes by form of pollen
(or by color of the flowers) are not quite exactly superimposed: they have the
same macroscopic architecture; more exactly the same architectural diversity.
But within each of these groups the principal subdivisions have their own areas;
with, in an intermediate position, the hybrids.
It is not the place here to set out even summarily the geometric principles of
multidimensional statistical analysis. Let us say again only that with a cloud, a
nebula, of individuals of a natural or other order described by a set of their properties (and not only by the measurements: it can be a matter of bacterial clones
described by traits of their metabolism ...) computer computation substitutes
plane maps, where gradations and qualitative oppositions, discontinuities that
subdivide the field studied, are inscribed. The computer is indeed the Novius
Organum of inductive definition.
8
Statistical Existence
Statistics is not a means of knowledge, draining into an idea the flood of facts;
it is a mode of being. Collective entities are not only constructions, even if
dynamic, of our spirit; they are defined in nature for an equilibrium, an exchange
of information between individuals. Here is the definition that the eminent
paleontologist G.G. Simpson proposes for species8 .
“That category cannot be naturally defined in terms of static pattern or
morphology but only in terms of dynamic, evolutionary, genetical, concepts and
relationships among and between populations ... Taxonomic studies are always
statistical in nature. The true object of enquiry, the population in nature, can
rarely be observed directly and entire ...”
Revealed by statistical study of a sample (cf. the Myosotis above) a species
is in itself a sample that never exhausts the set of diverse possible forms; if
the species is particularly few in number, or geographically broken up, it is
possible that from heredity or mutations the equilibrium of these possible forms
is unstable.
8 G.G.
Simpson, Principles of Animal Taxonomy, Columbia University Press (1961), p. 65.
10
The statistical definition of species as a system of masses (the individuals)
partially occupying a potential field (which is in the set, or more so in the
space, of possible living things) conflicts with the definition by a finite and
closed system of properties for which mathematical logic studies conjunction
and disjunction, writing formulas such as those presented above. Simpson,
whom the universe of the logical positivists does not attract, deplores that9
“Some present-day taxonomists advocate what is essentially a return to
scholastic principles.”
The term scholastic is ambiguous: for some, it serves to profess their fidelity
to the Doctors of whom the most illustrious is Saint Thomas Aquinas; with the
pen of others, this is a sign of contempt striking out what is however the so
diverse and often contradictory totality of the philosophical production of an
ignored epoch. Simpson, as for himself, is not well-versed in medieval thought.
He knows by hearsay that Bacon and Descartes denounced a very perfectioned
dialectic, but all the less capable of encompassing a real object. He notes that
the abuse of the formalism today places science in similar peril. We can, without
lacking respect to our masters of the 13th century, subscribe to these criticisms.
From the statistical mode of being, vegetal ecology offers a remarkable example; here again let us cite a classic: M. Guinochet10 writes:
“... Let us define in nature, for example on a lawn, a surface area of 1
m2 taken randomly, and let us note all species that one observes there. Let
us enlarge this surface area to 2 m2 . Our list lengthens by some species and
so on. Let us draw a plot, on which we have as abscissa the surface areas,
and as ordinate the corresponding numbers of species. We obtain a curve that,
after a more or less rapid climb becomes almost parallel to the abscissa axis.
Then if we continue to enlarge the surface area studied the curve will again
become more or less rapidly climbing. The surface area corresponding to the
first inflexion of the curve [we would rather say: at the start of the leveling
off, JPB] is the minimum area necessary for one to have a chance of finding
essentially all species of the “individual of association” studied; on the contrary
the surface area corresponding to the second inflexion of the curve [end of the
leveling off] is the maximum area that one can not go past without noting the
species foreign to the individual of association considered ...”.
To what natural reality does the individual of association correspond, the
floristically homogeneous surface area for which M. Guinochet has just offered
us an evidently very schematized definition? At a physico-chemical equilibrium,
indeed microclimatic where all species participate, that on the basis offered by
the minerals work out themselves the ground from which they live, in symbiosis
with micro-organisms. However the individual is not clearly circumscribed in the
species (a surface area of 15 to 40 m2 would be suitable according to Figure 2);
and it is also no longer logically defined by a set of characters – the species that
ought to be necessarily present there: a characteristic species can be missing
from the table, a foreigner one can be fortuitously inserted there. However from
9 Ibid.,
p. 24.
Guinochet, Logique et Dynamique du Peuplement Végétal (Logic and Dynamic of
Vegetal Populating), Masson, Paris (1955), p. 63.
10 M.
11
Figure 2: Plot of the readings serving to define an individual of association;
the floristically homogeneous area can be taken between about 15 and 40 m2 .
(Abscissa: surface area reading; ordinate: number of species.)
these individuals, already materially in themselves difficult to get hold of, whose
mode of being, hence a fortiori definition, subject to fluctuations are statistical,
the ecologist constructs a taxonomy: in the same was as the botanist says that
such a vegetal individual pertains to such a species, the ecologist recognizes
in diverse floristically homogeneous surface areas the same type (one says, the
same association) that is the logical analog of a species.
From the particular to the general, from the concrete to the abstract, from
the individual to the species, from the bunch of plants to the association (or
ecological type), this is always for the spirit the same ascending hierarchical
movement; a sort of passage to the limit causing degrees of knowledge to climb,
and also, in a certain sense, those of being. The mathematical imagination will
aid us in representing this passage.
9
Filter and Limit
The mathematical notion of filter which serves for the study of limits in the
most general spaces was introduced in 1937 by our Master H. Cartan. But
more so than to N. Bourbaki, we have recourse to a distant precursor, William
of Ockham, the Venerabilis Inceptor (Venerable Enterpriser), an English Friar
Minor who taught in Paris at the start of the 14th century. In the album of
philosophers the name of Ockham is illustrated by a razor: one calls Ockham’s
razor the principle according to which one should not without necessity postulate
beings: Entia non sunt multiplicanda praeter necessitatem.
Over his monumental history of cosmological doctrines11 , Duhem notes that:
11 P.
Duhem, Le Système du Monde (The System of the World), Hermann, Vol. VII, p. 236
12
“In accordance with this principle, Ockham uses for place the very definition of Aristotle, returning to it its fundamental simplicity, clearing it of every
parasitic addition: place, this is the part of the containing body which touches
immediately the contained body.”
This is the occassion for the Venerabilis Inceptor to employ the razor for a
fine topological dissection. P. Duhem translates thus Ockham:
“Place is that which is ultimate in the containing, that is, the ultimate part
of the containing body. It is not that there exists a certain ultimate part that is,
in its totality, distinct from other parts. I name ultimate part every part which
extends as far as the located body, which affects the located body in the place;
according to this manner of speaking, the ultimate part has itself a multitude
of parts that do not affect the located body.
But, you say, I take the ultimate part, that which, to begin with, is called
place; the former does not have certain parts that affect the located body and
others that do not affect it; otherwise this is not this part, but a part of this
part, which would receive in the first instance the name of place.
I respond that one must distinguish on the subject of the ultimate part.
In a first way, one names ultimate part every part that extends as far as the
located body and which affects immediately the contained body at the place;
in this sense, there is an infinity of ultimate parts that, all, are place. If an
ultimate part affects, by its right-hand side, the located body, that right half
of this tangent [literally: touching, JPB] part is still the ultimate part, and the
half of this half is also an ultimate part, and so to infinity.
In a second way, one names the ultimate part contiguous to the located body
that which is found after every other, contiguous to the located body, part. In
this latter sense, there is no part that is an ultimate part.”
In contemporary mathematical language a set F of non-empty parts of a
set E is called a filter (filter base) if given two of its parts, F and F  (F ∈
F, F  ∈ F), one can find a third F  that is contained at the same time in F and
F  (i.e. F  ∈ F; F  ⊂ F  , F  ⊂ F ). One recognizes a property that Ockham
attributes to ultimate parts: however closely one restricts an ultimate part, one
can restrict it more: under the name of ultimate parts, Ockham describes the
filter of neighborhoods of the contained in the containing (cf. Figure 3).
Let us call now, with P.S. Alexandrov, neighborhood of the infinite the
exterior of a bounded, but arbitrarily large, region of space: just as the location
of the contained is defined in the containing by a filter, in the same way the filter
of neighborhoods of the infinite defines what one will call, to follow Ockham,
the place of the infinite (cf. Figure 3). One can further adjoin an ideal place L
defined by a filter of ultimate parts F to a space E: the place L is not identified
with any part F ; every part F “has itself a multitude of parts that do not affect”
L and from which one can strip out F ; without however defining L completely
purely, because “In this last sense, there is no part that is the ultimate part”.
We consider that it is similarly that one can, at a given level of the real,
define as the limit place of a filter a being of higher hierarchical level.
et seq.
13
Figure 3: Two examples of filter. Left: the place of the infinite. Right: The
filter of neighborhoods of the contained. (Diagonal shaded area: the contained.)
10
Definition by a Filter
One said that one must not be proud of appearances: it is however according
to them that we know every thing. The first sight is often deceptive: but
more and more attentive considerations lead us quite close to the essence of the
object to deserve in the end to win our assent. In the half-light everything which
rustles appears living; and silence seems the sign of solitude or of death. Let us
stop: this noise was not the cracking of a branch under our step. A faint light
caresses a wing that one feels to quiver, that one believes to brush past. This
wing is a dry leaf that is stirred again in the air, that our arrival has shaken.
The exterior indications of life are neither necessary traits nor sufficient traits:
where life is present they can be missing, and their presence does not imply
that of life. Based on more profound characteristics, a sophisticated automat
will again deceive us. From qualities associated with life, only the ultimate part
allows a definite definition. One therefore conceives that an ideal limit notion –
to be living, to be good – is defined as comprehension by a filter on a set of the
qualities.
Rather than trying to decide on formulas, let us look at an example (cf.
Figure 4).
The set Q1 of the qualities that an individual is liable to possess is represented by the interior of a quadrant of a circle of radius 1. (We say interior so
that the vertex S does not represent a quality that would be perfection.) We
suppose that the more a quality is distant from S the less profound its contribution is to the limit notion L that we seek to define: for example, if L is life,
the faculty of emitting a sound spontaneously will be at the distance 0.9; while
the faculty of responding by a sound to a light will be at the distance 0.8. One
can postulate that an individual who possesses the set Qx of qualities situated
a distance less than x from S (x a positive number less than 1; Qx quadrant
of the circle of radius x) surely goes into the extension of the limit notion, L.
14
Figure 4: Shaded with diagonal lines is the domain of qualities that an individual
i possesses; it is convenient to attribute to i the limit notion L because, from
the qualities located at the distance x (on a quadrant of the circle) it possesses
a part greater than (1 − x).
But perhaps this is to make too few cases of qualities distant from S: even if
none among them, considered in particular, is indispensable to the notion L
(as to the notion of man, neither hearing nor sight are essential since there are
deaf and blind people), should not a certain part amongst them always remain?
One will say then for example that for an individual to possess the notion L he
should possess a part at least equal to 1 − x of the qualities situated at distance
x: that is to say, 10% of those situated at the distance x = 0.9; half of the those
situated at the distance 0.5; 90% of the those situated at the distance 0.1; etc.
We do not know if formal logic will come soon to treat as filters the systems
of embedded conditions that define a limit notion. Our Figure 3 is certainly
only a sketch which suffices however to show that logical properties of a limit
notion L are not those of a common predicate. We will clarify this in considering
negation.
If a notion, or predicate R, is defined by its extension it is not on another
plane than its negation 1 R . One has I, the set of all individuals; I(R ) = {i|i ∈
R (i) true }, extension of R (set of individuals that do not possess R and thus
by definition possess 1 R ). More independently of every model (that is, from
a point of view which is in a certain sense that of comprehension, cf. above in
section 6) one sees that if in the basis vocabulary of a theory the predicate R
appears it is possible to suppress R and to introduce R , which will have the
sense of 1 R and will be, in its place, a basic predicate (that is to say that in
the axioms of the theory one will replace R by 1 R ).
15
But while a part I(R ) of I is put opposite its complement I − I(R ) =
I(1 R ), a filter F (of parts of a set E) does not have a complement; (the set
Part(E) −F, the parts of E that are not in the filter F, is not a filter; the set
(E − F |F ∈ F) of complements in E of the F that are of F is not a filter either).
It is not possible to put opposite a limit notion L (defined as is suggested in
Figure 3, or in any other analogous fashion) a contrary notion, 1 L, liable to be
defined in the same way as L. In this sense a notion limit has not an opposite
that is on the same plane as itself.
This holds in particular for the notion of form. Thus “to have the form of
A” does not have an opposite: the set of patterns that are As is not defined with
respect to an ideal structure, to which any opposite structure does not respond,
which can define the “non-A”. In a printed text, the “non-A” is defined as:
“that which is B or C or D ... or Z ... or 9” but in general the set of patternless
objects is the inevitable context of every problem of pattern recognition. Life
defines itself; while death, indeed the inanimate, is only defined relative to life.
11
The Filter of the “secundum quid”
An aircraft flies. In the jet engines a continuous stream of kerosene is on fire;
through the nozzles a jet of burning gas escapes. Sometimes along the wing or
the center-board, a flap tilts, inclining with it the docile flight of the aircraft:
thus this flap, that a weak enough force suffices to move, is the master of the
aircraft whose mass is more than one hundred times greater. All however – flow
of fuel and coordinated interplay of flaps – is governed by the light levers which
obey the members of a man: this man, towards whom everything in the aircraft
converges, drives the flight.
What drives the man? Can one not within the human body, as within the
flying machine, climb the hierarchy of a filter? Let us take away integument
and carcass, intake of air and food, rate of oxygen and of sugar, pivots and
springs; because we know like Alexandrov that what we are seeking is beyond
these organs and these bounded functions. What remains: some relays where
information that flows is reverberating, coded as impulsions, as electrochemical
waves, as molecules, perhaps articulated; the brain! Without doubt; but, from
the vegetative stream that we thought at first to have completely discarded,
relay and information are not absolutely separated: the neurons are, on the
contrary, the seat of a metabolism of the most active. It does not suffice to
carve up in the space of the body a filter of ultimate parts closing in around
that which we seek: it must be filtered on site; to distinguish, within a single
apparatus, levels that spatially coexist; separating that which appears lower
and material so as to keep only that which formally controls it; to dissect the
ultimate parts by circumscribing them, not by a volume but by a definition:
Fn+1 , this is Fn considered in how it is active and formal; this is Fn restrained
according to some condition, Fn secundum quid as would have said a Doctor of
the century of Saint Louis.
In the end, from condition to condition, one realizes that everything balances
16
on an evanescent interplay of matter and energy. The soul is not at the end of
a scalpel; it is at the end of Ockham’s razor.
17